On generalized square-full numbers in an arithmetic progression

Let a and b∈ℕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \in \mathbb{N}$$\end{document}. Denote by Ra,b the set of all integers n > 1 whose canonical prime representation n=p1α1p2α2…prαr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = p_1^{{\alpha _1}}p_2^{{\alpha _2}} \ldots p_r^{{\alpha _r}}$$\end{document} has all exponents αi (1 ⩽ i ⩽ r) being a multiple of a or belonging to the arithmetic progression at + b, t∈ℕ0:=ℕ∪{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in {\mathbb{N}_0}: =\mathbb{N} \cup \{ 0\} $$\end{document}. All integers in Ra,b are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given.